论文标题
简短细分的尺寸尺寸
The size-Ramsey number of short subdivisions
论文作者
论文摘要
图$ h $的$ r $ -size-size-ramsey number $ \ hat {r} _r（h）$是图$ g $可以具有的最小边缘，因此，对于$ r $ colors的每种边缘颜色$ r $ colors的每种边缘都存在$ h $ in $ g $中的$ h $。对于图$ h $，我们用$ h^q $表示，从$ h $获得的图形通过用$ q { - } 1 $ vertices对其边缘进行了细分。在最近的kohayakawa，crotter和r {Ö} dl的论文中，结果表明，对于所有常数整数$ q，r \ geq 2 $以及$ n $ pertices上的每张图$ h $ in $ n $ pertices和有限的最高学位，$ r $ r $ size-size-size-size-size-size-ramsey $ h^q $的数量最多是$ h^q $，最多是$（\ log n）^$ N） $ n $大。我们通过证明任何此类图$ h $的$ \ hat {r} _r（h^q）\ leq o（n^{1+1/q}）$来改进该结果。
The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest number of edges a graph $G$ can have, such that for every edge-coloring of $G$ with $r$ colors there exists a monochromatic copy of $H$ in $G$. For a graph $H$, we denote by $H^q$ the graph obtained from $H$ by subdividing its edges with $q{-}1$ vertices each. In a recent paper of Kohayakawa, Retter and R{ö}dl, it is shown that for all constant integers $q,r\geq 2$ and every graph $H$ on $n$ vertices and of bounded maximum degree, the $r$-size-Ramsey number of $H^q$ is at most $(\log n)^{20(q-1)}n^{1+1/q}$, for $n$ large enough. We improve upon this result using a significantly shorter argument by showing that $\hat{R}_r(H^q)\leq O(n^{1+1/q})$ for any such graph $H$.
